Detail Geometric analysis of delayed bifurcations Bibliografi Author: Hayes, Michael George ; Kaper, Tasso J. (Advisor); Kopell, Nancy (Advisor) Topik: MATHEMATICS Bahasa: (EN )    ISBN: 0-599-44208-5     Penerbit: Boston University     Tahun Terbit: 2000     Jenis: Theses - Dissertation Fulltext: 9942385.pdf (0.0B; 0 download) Abstract Delayed bifurcations play a central role in differential equations modeling a variety of biological, chemical and physical processes. This phenomenon occurs in analytic systems of differential equations which are dependent on a slowly varying parameter. Solutions remain close to an unstable equilibrium point, much past the bifurcation value at which the equilibrium point loses stability. This apparent “delay” in the effect of the bifurcation is anti-intuitive, in the sense that one would normally expect solutions to be repelled from the equilibrium point as soon as it turns unstable. More importantly, a maximal length of delay, or “buffer point”, often exists. The delayed bifurcation phenomenon is also part of a wider class of problems in geometric singular perturbation theory. This theory treats normally hyperbolic center (or “slow”) manifolds. In the context of delayed bifurcation problems, it establishes the existence of one-parameter families of such manifolds to the left and right of, but not in the immediate vicinity of, the bifurcation point. The crucial question is: what is the relative configuration of these two families of center manifolds? A method is developed in this dissertation to geometrically analyze the delayed passage problem. This method is first developed on the linear Shishkova equations. The theory is then extended to a class of nonlinear problems that includes the ubiquitous delayed Hopf bifurcation example. We track these two families of manifolds along paths in the complex time plane on which the behavior of the differential equations is purely hyperbolic (and where the method of geometric desingularization may be applied). The choice of paths avoids the complications associated with the mixed hyperbolic—elliptic behavior of the differential equations along the real time axis. From that, we show that, under certain conditions, the families of center manifolds are separated by an exponentially small amount, and that there exist finite buffer points. Counter examples are provided to show that these conditions are necessary. Opini Anda Klik untuk menuliskan opini Anda tentang koleksi ini! Lihat Sejarah Pengadaan  Konversi Metadata   Kembali

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